### Unit 1: Preview and Review

While your first course in calculus can strike you as something new to learn, it is not comparable to learning a foreign language where everything seems different. Calculus still depends on most of the things you learned in algebra, and the true genius of Isaac Newton was to realize that he could get answers for this something new by relying on simple and known things like graphs, geometry, and algebra. There is a need to review those concepts in this unit, where a graph can reinforce the adage that a picture is worth one thousand words. This unit starts right off with one of the most important steps in mastering problem solving: Have a clear and precise statement of what the problem really is about.

**Completing this unit should take you approximately 6 hours.**

Upon successful completion of this unit, you will be able to:

- approximate a slope of a tangent line from a function given as a graph;
- approximate the area of an irregular figure by counting inside squares;
- calculate the slope of the line through two points;
- write the equation of the line through two points using both slope-intercept and point-slope forms;
- write the equation of a circle with a given center and radius;
- write the equation of a line through a point given the slope;
- write the equations of lines that are parallel or perpendicular to a given line;
- evaluate a function at a point, given by a formula, graph, table, or words;
- evaluate a combination, or a composition, of functions when indicated by the symbols , , , and ;
- evaluate and graph the elementary functions as well as and ;
- state whether a given "if-then" statement is true or false, and justify the answer;
- state which parts of a mathematical statement are assumptions or hypotheses, and which are conclusions;
- state the contrapositive form of an "if-then" statement; and
- write negative statements.

### 1.1: Preview of Calculus

This section reviews how algebra will help you in the study of calculus. In this unit, you will review two main topics: finding the slope of tangent lines from a graph and finding the area of an irregular shape.

Read this section for an introduction to calculus.

Watch this video (until 8:12) on how to find the slopes of tangent lines from a graph.

Watch this video on how to find the area of an irregular shape.

Work through the odd-numbered problems 1-7. Once you have completed the problem set, check your answers.

### 1.2: Lines in the Plane

In this section, you will begin the study of calculus. It is important to understand the properties and equations of straight lines since they set the tone for differential calculus.

Read this section and work through practice problems 1-9.

Watch this video on how to find the slope of a line given two points on a graph.

Watch this video on how to write the equation of a line through two points using the slope-point formula, and converting the equation to slope-intersect form.

Watch this video on how to write an equation of a circle with the given center and radius.

Watch this video on how to write an equation of a line through a point given the slope.

Watch this video on how to find the equation of a line going through a point and parallel to another line.

Watch this video on how to find the equation of a line going through a point and perpendicular to another line.

Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

### 1.3: Functions and Their Graphs

Functions and their graphs are essential concepts in calculus. You will learn about the behavior of functions, which are ideas that you can apply to many fields.

Read this section for an introduction to functions and their graphs. Work through practice problems 1-5.

Watch this video on evaluating function at a point.

Work through the odd-numbered problems 1-23. Once you have completed the problem set, check your answers.

### 1.4: Combinations of Functions

In this section, you will continue to learn about functions. You will take it a step further by combining basic functions to solve more complicated problems in calculus.

Read this section for an introduction to combinations of functions, then work through practice problems 1-9.

Watch the first 16 minutes of this video on evaluating composition of functions.

Watch this video on combining functions using algebraic operations.

Watch this video on evaluating functions given equations, graphs and tables.

Work through the odd-numbered problems 1-31. Once you have completed the problem set, check your answers.

### 1.5: Mathematical Language

This section is very important in learning calculus. It is important to understand some common mathematical phrases to be able to understand the concepts being discussed.

Read this section for an introduction to mathematical language, then work through practice problems 1-4.

Watch this video on how to negate a quantified statement.

Watch this video on converse and contrapositive of a conditional statement and how to write them.

Work through the odd-numbered problems 1-25. Once you have completed the problem set, check your answers for the odd-numbered questions.